Computation of Smallest Eigenvalues using Spectral Schur Complements

The Automated Multilevel Substructing method (AMLS) was recently presented as an alternative to well-established methods for computing eigenvalues of large matrices in the context of structural engi-neering. This technique is based on exploiting a high level of dimensional reduction via domain decom-position and projection methods. This paper takes a purely algebraic look at the method and explains that it can be viewed as a technique based on a first order approximation to a nonlinear eigenvalue problem. A ‘corrective projection ’ viewpoint leads us to explore variants of the method which use Krylov sub-spaces instead of eigenbasis to construct subspaces of approximants. The nonlinear eigenvalue problem viewpoint yields a second order approximation as an enhancement to the first order technique inherent to AMLS. Numerical experiments are presented to validate the approaches presented.